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3 years ago

How do you factor a²+2a+2 using imaginary numbers?

Mathematics

1 answer:

katovenus [111]3 years ago

5 0

Answer:

Step-by-step explanation:

a²+2a+2

$4^{2} +2(4)+2$

16+8+2

24+2

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3. The curve C with equation y=f(x) is such that, dy/dx = 3x^2 + 4x +k

Andreas93 [3]

a. Given that y = f(x) and f(0) = -2, by the fundamental theorem of calculus we have

$\displaystyle \frac{dy}{dx} = 3x^2 + 4x + k \implies y = f(0) + \int_0^x (3t^2+4t+k) \, dt$

Evaluate the integral to solve for y :

$\displaystyle y = -2 + \int_0^x (3t^2+4t+k) \, dt$

$\displaystyle y = -2 + (t^3+2t^2+kt)\bigg|_0^x$

$\displaystyle y = x^3+2x^2+kx - 2$

Use the other known value, f(2) = 18, to solve for k :

$18 = 2^3 + 2\times2^2+2k - 2 \implies \boxed{k = 2}$

Then the curve C has equation

$\boxed{y = x^3 + 2x^2 + 2x - 2}$

b. Any tangent to the curve C at a point (a, f(a)) has slope equal to the derivative of y at that point:

$\dfrac{dy}{dx}\bigg|_{x=a} = 3a^2 + 4a + 2$

The slope of the given tangent line $y=x-2$ is 1. Solve for a :

$3a^2 + 4a + 2 = 1 \implies 3a^2 + 4a + 1 = (3a+1)(a+1)=0 \implies a = -\dfrac13 \text{ or }a = -1$

so we know there exists a tangent to C with slope 1. When x = -1/3, we have y = f(-1/3) = -67/27; when x = -1, we have y = f(-1) = -3. This means the tangent line must meet C at either (-1/3, -67/27) or (-1, -3).

Decide which of these points is correct:

$x - 2 = x^3 + 2x^2 + 2x - 2 \implies x^3 + 2x^2 + x = x(x+1)^2=0 \implies x=0 \text{ or } x = -1$

So, the point of contact between the tangent line and C is (-1, -3).

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2 years ago

Hector gather data from eight students who have accounts with to social networking apps about the number of people they connect

Natasha_Volkova [10]

hi gig hi hi hi hi sorry :answer eight networking connect hector students

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2 years ago

Show how to work the division problem 47 divided by 8 please

Dvinal [7]

Answer:

Step-by-step explanation:

47 divided by 8

= 5.875

hope this helps

6 0

2 years ago

Y=x^(2) -2x - 3 and y = x - 3

Oliga [24]

Answer:

(0, -3) and (3, 0)

Explanation:

1st equation: y = x² -2x - 3

2nd equation : y = x - 3

Using substitution method:

x² -2x - 3 = x - 3 exchange sides

x² - 2x - x - 3 + 3 = 0 simplify

x² - 3x = 0 take common factor

x(x - 3) = 0 simplify

x = 0, 3

Solve for y:

when x is 0, y = x - 3 = 0 - 3 = -3

when x is 3, y = x - 3 = 3 - 3 = 0

5 0

2 years ago

Sum of series 1 +1/3 +1/3².... is?

Charra [1.4K]

Answer:

The sum of the series is 3/2

Step-by-step explanation:

Given

1 + 1/3 + 1/3^2 + ....

Required

The sum of the series

This implies that we calculate the sum to infinity.

We have:

$a = 1$ -- The first term

First, calculate the common ratio (r)

$r = \frac{1}{3^2} / \frac{1}{3}$

Change to product

$r = \frac{1}{3^2} * \frac{3}{1}$

Solve

$r = \frac{1}{3}$

The sum of the series is then calculated as:

$S_{\infty} = \frac{a}{1 - r}$

$S_{\infty} = \frac{1}{1 - 1/3}$

Solve the denominator

$S_{\infty} = \frac{1}{2/3}$

Express as product

$S_{\infty} = 1 * \frac{3}{2}$

$S_{\infty} = \frac{3}{2}$

7 0

3 years ago